THE EFFECT OF INSIDER TRADING BY A DOMINANT
TRADER IN A SIMPLE SECURITIES MARKET MODEL
by
Sumru A1tug
Discussion Paper No. 212, March, 1985
First Draft August, 1982
Revised December, 1984
Center for Economic Research
Department of Economics
University of Minnesota
Minneapolis, Minnesota 55455
Abstract
This paper studies the effects of strategic behavior by an informed trader
who is large relative to the remaining traders in a simple securities market
model.
The strategic interaction among market participants is modelled as a
price-setting game in which the dominant trader sets price, taking into account
the reactions of uninformed traders who behave competitively.
The equilibrium
outcomes of this game are compared with those which emerge when all traders act
as price-takers and/or all information is public.
Besides characterizing the
informational efficiency of prices under alternative assumptions and describing
the dual advantages conferred by superior information and the ability to move
first and set prices, these comparisons help to determine whether
o~~~rvations
on equilibrium prices and quantities traded suffice to reveal the existence of.
strategic insider trading in an otherwise competitive market.
1.
Introduction
That insider trading is an important phenomenon in securities markets
emerges from numerous studies, including those by 8aesel and Stein [1979],
Finnerty [1976], Jaffe [1974], and Pratt and DeVere [1970].
Furthermore, there
exists empirical evidence to suggest that infonmed traders may influence the
price of securities with their trades or offers to trade.
For example, the
studies by Dean, Mayers, and Raab [1977], Grier and Albin [1973], Kraus and
Stoll [1982], and Raab [1976] show that large blocks of stock and secondary
offerings trade at prices significantly different from trades executed only
minutes earlier. Moreover, the view that emerges from these studies is that
such price changes reflect new information revealed through the offer to trade.
In a different context, the belief that informed traders
can~affect
security
~
prices forms the basls for various regulatory measures designed to control
insider
trading~
As an example, Krynasowski [1979] notes that sudden jumps in
the price of a particular security have, on occasion, been interpreted as signalling the existence of insider trading and led to trading suspensions in that
security, presumably in an effort to publicize the "monopoly" information.
Motivated by the empirical findings cited above, this paper analyzes the
effect of strategic behavior by an informed trader who is large relative -to
the market.
The strategic interaction among market participants takes a par-
ticularly simple form:
conditional on observing a noisy signal about future
payoffs, an informed trader sets the
p~ice
of a particular security, taking into
account the. reactions of uninformed traders who behave
competi~ively.
.
As a
description of actual behavior in securities markets, the price-setting assumption may be appropriate for an insider who trades in large blocks or who stands
ready to buy and sell sizable quantities at a given. price (as, for example,
during merger or takeover bids).
At the same time that it allows a simple
-2departure from price-taking behavior for . infonmed traders, the assumption
that
.
the insider moves first and sets prices provides a price formation mechanism
which is consistent with uninformed traders using observations on price to compute their demands and to make inferences about the inside information.
The equilibrium concept used to derive the optimal strategies of the informed and uninformed traders for the price-setting game differentiates this
paper from some recent papers dealing with the interaction between information
and market power. One such paper is by Kihlstrom and Postlwaite [1983J who
show that if a monopolist can pre-commit to a given strategy prior to receiving
an informative signal about the future state, then he may be better off, !!
ante, by ignoring his information. This is because using his information will
cause it to be Fevealed through the equilibrium
price~and
thereby
eli~inate~x
ante risk sharing opportunities for both the monopolist arid the competitive
trader.
These authors.also describe the role of random strategies as an optimal
form of concealment which occurs when the monopolist pre-commits to basing his
pricing decisions on a less informative signal than the one he actually receives.
Second, using a framework similar to the one in this paper, Grinblatt and Ross [1983]
examine the effect of market power possessed by a large informed risk-neutral
trader with a linear demand schedule. However, while they characterize the best
linear demand schedule for the monopollst, they do not demonstrate that demand
functions which are linear in the monopolist's information and the current price
of the asset constitute an optimal strategy.
furthermore, they assume that the
monopolist will not deyiate from the actions prescribed .. by ~is decision rule
once he knows the value of his 'signal and the price
~f
the asset.
In fact, one
possible deviation for the monopolist which always guarantees non-negative ex
post profits is to buy (sell) a~bitrarily lar~e quantities of the asset w~enever
the expected value of the future payoff, conditional on the monopolist'S signal
-3-
exceeds (falls short of) the current price of the asset.
By contrast, this
paper uses" an equilibrium concept" which is subgame perfect.
Consequently, it
tries to analyze the effects of strategic trading by an informed trader in an
otherwise competitive market when pre-commitment on the part of this trader cannot be enforced.
Section 3 characterizes the equilibrium outcomes with an insider who behaves strategically, setting prices in a simple two-period, two asset economy
with a continuum of uniformed traders. To separate out the effects of superior
information and the ability to move first and set prices,this section also
describes the equilibrium under alternative assumptions.
First, the rational expectations equilibrium of Grossman and Stiglitz
[1976] with informed and
, uninformed traders
against which to compare the
effe~ts
i~used
as a natural
benchmark
,
of non-competitive behavior by tbe insider.
Provided each uninformed trader is unsure about the initial position or endowments of the remaining traders, it turns out that the price set by the insider
as well as the rational expectations equilibrium price when all traders behave
competitively conveys only statistical information about the noisy signal. This
occurs because changes in supply become confounded with changes in information,
preventing the full revelation of information. However, compared to the case
when all traders act as price-takers, the optimal price-setting rule implies
that price will respond less to any given piece of news received by the insider.
In effect, the insider increases his expected utility (or expected profit) by
making price less correlated with his information.
To examine further the role
o~
informational advantage;" Section 3 also
describes the equilibrium outcomes when all information is public but a large
trader (who may be interpreted asa specialist or market maker) sets prices.
Elimination of the informational monopoly "implies" that all profit opportunities
-4due to information are exhausted by the trades of the competitive fringe.
With-
out a private source of information, the dominant trader merely behaves as a
textbook monopolist (monopsonist) facing an exogenous demand (supply) curve.
Alternatively, differential infonnation confers greater market power on the
dominant trader because the demands of uninformed traders are jointly determined
with the equilibrium price.
By optimally setting price as a function of his
infonnation, the informed monopolist can alter the beliefs of uninfonned traders
and thereby increase his profits.
The possibility that superior infonnation alone confers market power on
individual traders has also been recognized by various authors. 2 Commentina
on the literature dealing with infonnation revelation and aggregation in
securities markets--exemplifi!d ~y the~Work of Grossman [1976], [1978], Grossman
and Stiglitz [1976], Danthine [1978], Bray [~981]; Verrechia [1982]--He11wig
[1980] points out that the practice
~f
modelling infonned traders as price-
takers leads to a "schizophrenia" problem. Although these traders are aware of
the covariance between their own infonnation and price, they do not attempt to
"man ipu1ate the price or the information content of price".
His solution is to
consider a limit economy where the equilibrium price still aggregates all
avai~
able information but the effect of each individual trader becomes negligible.
A recent paper by Kyle [1984] proposes a different solution:
he considers a
model with \lnoise traders and a large number of infonned and uninfonned specull
1ators who take into account the effect of their trades on
Characterizing a Nash
equi1ibri~m
~qui1ibrium
price.
in whJch each trader takes as given the strat-
egies of the remaining traders and conditions on his private information and the
equilibrium price when choosing his demand function, Kyle [1984] shows that some
of the undesirable
librium disappear:
c~nsequences
of the competitive
ration~l
expectations equi-
even in the absence of noise trading, prices reveal only
-5half the precision of each trader's information.
When noise trading exists, the
incentives for costly information acquisition are greater because price incorporates a smaller amount of the private information compared to the competitive
case.
Finally. a well-defined equilibrium exists even if speculators are risk-
neutral.
Similar results emerge from my price-setting game with inside informa-
The reason is that the equilibrium outcomes of the price-setting game also
tion.
describe those of a quantity-setting game similar to Kyle's where the insider
chooses asset demands based on an observation of the price, his private information and the value of his initial endowment.
However, even under the second
interpretation, there exist differences between
~le's
[1984] framework and mine.
First, uninformed traders behave as price-takers in my model:
it is only the
insider who exploits the joint 'advantages of his informational· monopoly and his
...
size relative to the market.
t
..
Furthermore, all traders are rational •. In other
words, there are no "noise" traders in·my framework who purchase a random, inelastic quantity regardless of their losses.
Finally, this paper addresses
some issues not considered by Kyle [1984] and others.
In particular, it tries
to determine whether observations on equilibrium prices and quantities traded
suffice to identify the existence of strategic insider trading in an otherwise
competitive market.
For example, it shows that contrary to some commonly held
beliefs, insider trading does not necessarily lead to "excess" variability of
. equilibrium prices:
by taking into account uninformed traders' reactions, the
insider who behaves strategically may reduce the variance in price. compared to
the competitive outcome.
The rest of this paper is organized as follows.
model.
Section 2 describes the
Section 3 presents the equilibrium under different information structure
and assumptions about the behavior of the large trader.
Equilibrium outcomes are
-6-
analyzed in Section 4.
Section 5 discusses whether such market outcomes can be
identified from observable series on prices and quantities traded.
-72.
The Model
Consider a simple two period economy:
in the first period, individuals can
trade claims to two assets, a risky asset which has an uncertain payoff of r
and a riskless asset which yields one unit with certainty. The market at date
one in which these claims are traded is comprised of a continuum of uninformed
traders and an informed trader or insider who receives a noisy signal y
about the future payoff r.
=r
+ v
The two types of traders are also differentiated
with respect to size: while uninformed traders are non-atomic, the insider is
large relative to the market in the sense that he is of positive measure, normalized here at one.
Furthermore, each uninformed trader is assumed to be risk-
averse with a strictly positive coefficient of absolute risk aversion equal to
au.
The insider mayor may not be risk-averse, depending on the context.
Hence, aI
~
O.
The initial wealth of each trader is determined by his initial position or
endowment of the two assets.
Uninformed traders, who are identical in prefer-
ences and beliefs, also possess the same number of shares of the riskless asset,
denoted by Bui
= -B.
However, they differ with respect to their ownership of the
risky asset:
a fraction m has a large initial position in the risky asset,
denoted by e iu = e1u while the remainder l-m own only e ui = e2u shares with
eu1 > .2
eu , 0 < m < 1 • Similarly, the insider's initial wealth is determined by his
endowment of the two assets, denoted by Bl
and e , respectively. The
l
initial wealth of each trader fluctuates randomly because his initial holdings
of the risky asset fluctuate
randomly, i.e., e ul , e2u and el are realizations
.
from random variables. On the other hand, the values of Band Bl are fixed
and known by all traders.
In period one, all traders evaluate the uncertainty in their environment
with the same prior beliefs, which specify that the random variables
r, v, el
-8-
and e~ and e~ are independently and normally distributed with means rm, 0,
m 1m
2m
.
2
2
2
2
2
eI' eu and eu and var1ances or' 0v' 01' 0lu and 02u. Furthermore,
the structure of these beliefs is common knowledge.
Here l/o~ measures the precision of the insider's signal, o~ the riskiness of investment opportunities, o~ the variability of the insider's initial
wealth and m the distribution of wealth among uninformed traders.
Finally,
and
(2.2)
2
0u
2
= m20lu
22
+ (l-m) 02u
determine distri bution of uninformed· traders' aggregate holdings of the ri sky
.
asset, defined by the normal random variable
(2.3)
'u =
J: .~
= me~
+
dp(l) +
J~ .~ dp(l)
(l-m)e~.
In this definition, p(i) denotes the Lebesgue measure on the unit interval.
The information possessed by each trader in this economy depends on both
public and private sources of information.
Prior to trading, the insider re-
ceives a piece of news y about the future payoff r.
Each trader also ob-
serves the number of shares of the risky asset he is endowed with in period one.
Further, all traders learn the total number of shares available for trade in the
risky asset. "From this information, the insider also infers uninformed traders'
aggregate endowment. i.e., knowledge of the total
e
= eI
+
me~
+
(l-m)e~ to-
gether with e I allows him to infer the current realization of eu• On the
other hand, any given uninformed trader observes only e - me~ = eI + (l-m)e~,
-9-
i,j
= 1,2, i ; j and is unable to infer the value of eI or eu.
~.
~10~
3.
Eguilibriwn
The large trader of this model derives market power from his informational
advantage as well as from his size relative to the market.
This section describes
the securities market equilibrium which emerges when the insider optimally exploits both sources of market power.
As a benchmark with which to compare the con-
sequences of strategic behavior with inside information, it also describes the
equilibrium derived under the assumption that information is private but the informed trader nevertheless behaves competitively.
However, even if the large
trader loses his informational advantage, his size relative to the remaining
trader still enables him to affect equilibrium outcomes.
Consequently, a second
benchmark is provided by the dominant trader equilibrium with public information.
The way in which the insider makes use of his dual advantages is simple:
conditional on observing y,
eI , and e or e u' the informed trader sets a
price' p for the risky asset to maximize the expected utility of next period's
wealth WI
= rX I
+ BI
and agrees to fulfill all desired trades at this price:
Consequently, his position in the market XI
XI
= eI
is determined by the constraint
l _ Xl + e2 _ x2
u
u
u
+ eu
1 - 1
2
2
where e u - Xu and e u - Xu denote the excess demands of "wealthy" and "poor"
uninformed traders. On the other hand, each uninformed trader behaves competitively.
Taking as given the price set by insiders, he maximizes his expected
utility of wealth W~
= rx~ ~ B~
by choosing how many shares X~ and B~ of
,the risky and riskless asset to hold.
engage in short sales of the assets.
Both types of traders are permitted to
In period two, the payoff on the risky
asset is realized and consumption takes place.
When setting a price, the insider recognizes the effect of his decision on
the demands of the uninformed traders and takes their reaction into account.
-11-
While the insider observes the total endowment eu of the uninformed traders,
he must nevertheless conjecture a set of demand functions X~(p) and B~(P) in
order to determine their net trades at
~
price p.
In this way, he also
determines the value of his own position for different values of the price.
Similarly, when computing their demands, uninformed traders use observations on
price to update their prior beliefs about the uncertain payoff r.
To do this,
they must conjecture a price-setting rule p(y,eI,e U) which specifies the price
charged by the insider for
Defining p: R3 ~ R,
realization of y, eI and e u.
X~: R ~ R and B~: R ~ R as the strategies for the
~
insider and each uninformed trader, respectively, the strategies p*(y,eI,e u)'
.*
.*
X~ (p) and B~ (p) constitute an equilibrium if each is a best response to the
.~
(p*(y,eI ,eu)~ X~*(p) B~*(P) for all
other. More formally,
i E [O,lJ) is
an egui1ibrium if
subject to px ui + Biu -< pe iu + Bu
(3.2)
p(y,eI,e u) E argmax E1[-exp(-aI(rX + BI»ly,e ,e,x~*(p),B~*(P)]
1
I
p
subject to pX 1 + BI
XI
= eI
~
pel + BI
1
"
"*
+ 0 (e~ - X~ (p»d~(i).
J
Notice that in (3.1), each uninformed trader's expectation about the future payoff r is conditional on observations of his individual endowment e~, the aggregate e,
price p and the conjectured equilibrium strategy p*(y,eI,e U )
for the insider. Similarly, because the insider always holds the residual position in the market as shown by the second constraint in (3.2)4, his pricing
-12decision depends on his conjecture about the uninformed traders' demands.
The
equilibrium pricing rule and asset demands are such that each trader's conjecture about other traders' behavior is confirmed.
Notice that in this definition of equilibrium, the dominant trader makes
his pricing decision after observing the current realization of the signal and
endowments.
Consequently, unlike the papers by Kih1strom and Postlewaite [1983J,
and Grinb1att and Ross [1982J, issues of pre-commitment do not arise.
Altern-
atively, the equilibrium of this paper is subgame perfect, with the postulated
strategy constituting a Nash equilibrium for subgfames defined for
~
realiza-
tions of y,
e I and eu '
Second, note that the large trader also serves as an actual price-setter
in this market, satisfying the demands of uninformed traders in order to keep
price at the value defined by p = p(y,eI,e u)'
An alternative way of defining
equilibrium is to assume that, conditional on information available in period
one, all traders choose excess demand schedules showing the quantity demanded as
a function of the current price and other relevant variables and submit these
schedules to a fictitious auctioneer.
The equilibrium price is then chosen by
the auctioneer to make the value of these excess demands equal to zero.
If the
large trader chooses an excess demand schedule which is perfectly elastic at the
price defined in (3.2), then the two interpretations of equilibrium lead to
identical outcomes.
In this case, the insider effectively sets price by trading
only at the price defined by p(y,eI,e u )' However, under either interpretation,
the set of possible strategies available to the traders is very large. Considering again the case of the price-setting insider, one alternative to the
strategy of satisfying all excess demands of uninformed traders at the price
p(y,eI,e ) is to charge a price subject to a limit on short sales, i.e., choose
u
a price p to satisfy (3.2) with the additional constraint that xI ~ -B.
-13Another possible strategy is to price successive blocks of stock. Despite the
potential interest. finding and characterizing the optimal strategy in such general strategy spaces is a diff4cu1t task and not attempted in this paper.
In what follows, I further restrict the strategies in (3.1) and (3.2) to be
linear in the relevant variables. Given the extensive literature which has
analyzed linear rational expectations equilibria in similar models. such a
restriction does not seem to be a serious shortcoming in this context as well.
This implies a price-setting rule linear in the infonned traders' infonnation y
and the values of initial endowments e l and eu' Similarly. the demand schedules of uninfonned traders are restricted to be linear in price. Since uninformed traders are identical in all respects except for the value of their
fnitial endowments. their equilibrium demand functions x~"* (p) are conjectured
to be identical.
In this case, the functions B~*(P). i E [0.1] showing un-
informed traders' demand for the riskless asset will differ across traders only
. because of their dependence on the value of e~. Under these conditions. I can
prove
Proposition 3.1: The set of coefficients (n O,n 1,n 2,n3 ,QO,Ql) describe a linear
price-setting equilibrium with inside information such that
p*{y,el,e u)
and
= nO
+ n1 Y + n2 e l + n3 eu
-14[_ QO (1 _ D ) + o2 r m/(o2 + o2}]
Q
v
r
v
2
1
2
or
and
Furthennore,
1
+ 2a 1
Proof:
<
'11'2
<
0 a nd
'II'
3
<
O.
See Appendix I.
To identify the consequences of informed trading by a dominant trader, I
·also prove
Corollary 3:
The set of coefficients
(nO,nl,n2,n3'~O'~1'~2)
price-setting equilibrium with public information such that
describe a
-15-
and
+ a )
u
+ 2a u )
Proof:
Foll~s
directly from the proof of Proposition 3.1 by noting that
• observing price yields no
add~tional
information to the competitive fringe when
y is public information.
The assumption that the insider sets prices by-taking into account the
reactions of uninformed traders may appear as an extreme description of behavior
in securities markets.
However. it turns out that the outcomes implied by
Proposition 3.1 or its corollary are also consistent with a quantity-setting
game in which the large trader chooses a linear demand function. taking into
account his effect on the equilibrium price and using observations on the price
'p,
his private information y and the value of his endowment e I • The uninformed traders behave competitively and choose asset demands as a linear function of price.
The.. equilibrium price is determined from the market-clearing
condition which sets the sum of the excess demands of informed and uninformed
traders equal to zero.
moves
fi~st
Unlike the price-setting game where the large trader
and announces a price at
~hich
he is willing to fulfill all trades,
the quantity-setting game is a simultaneous move game where all traders choose
~
-16-
asset demand functions.
To see how the large trader takes into account his effect on the equilibrium
price, notice that the price may be expressed as a function of XI
by using the
market-clearing condition
0*
and conjecturing that X~ (p)
(3.3)
p
a
1
= - --O + --
al
al
= aO
+ alP
1
al
with al ; 0,
i.e.,
(e +e ) - -- X •
I u
I
It is easy to show that the coefficients in the price-setting rule of
Proposition 3.1 or its corollary also describe the equilibrium price for the
simultaneous move game when the large trader chooses linear asset demands XI ~
and BI
to maximize his expected utility of end-of-period wealth WI
subject
to his budget constraint and given condition (3.3). As before, the asset demands
of the uninformed traders,
X~(p) and B~(P)' solve (3.1). The two different
games yield identical equilibrium outcomes due to the existence of a single
large trader who behaves non-competitively.
In the absence of any restrictions
on short sales, choosing asset demands XI and BI to satisfy (3.3) and the
single period budet constraint is equivalent to choosing price such the XI and
BI are determined from
and
Alternatively, the equilibrium outcomes of the price-setting game will not
be identical to those of the quantity-setting game when there exist several large
-17-
traders who behave as imperfect competitors. However, for the purposes of this
paper, the ability to interpret the equilibrium outcomes of Proposition 3.1 and
its corollary in tenms of the quantity-setting game has one advantage: that is,
I can use the competitive rational expectations equilibrium as a natural bench-
mark for analyizing the consequences of non-competitive behavior by the insider.
In this case, the informed trader bases his decisions on his private
infor~
mation but otherwise behaves competitively. Furthermore, all traders move
simultaneously and,taking prices as given, choose the quantities they wish to·
trade at these prices. However, in order to do this, they must conjecture a
pricing function which describes the statistical relationship between price, on
the one hand, and the signal y and endowments e I +
Defining the functions p: R2 + R, XIl R2 + R,
eu' .on the other.
BI : R2 + R, X~:~R + R
and B~: R+ R, the pricing function p*(y,e I + eu) and asset demands x~*(p), - ..
Bui* (p), i E [0,1], Xi ( p,y) and Bi(p,y) consti~ute an equilibrium if
(3.4)
x~*(p), B~*(P)
E
a~m~x E~[-exp(-au(rx~
1
+
B~»le~,e,p,p(y,eI
+ e )]
1
u
Xu·Bu
subject to pxi + Bi < pe i + Si
u
uu
u
l
.
J e~dp(i)
+ 0
(3.6 )
The first part of this definition merely describes how each uninformed
. trader maximizes the expected value of next period's wealth, conditional on the
value of his endowment and the aggregate e as well as on observations of the
-18-
current price and his conjecture about the equilibrium pricing function
p(y.eI+e u}. The analogous problem for the informed trader is presented in (3.5).
The next proposition characterizes the pricing function which is linear in the
insider's signal and the initial endowments such that for any realization of y.
e and eu' markets clear according to Condition (3.6).
(oO.ol.o2'YO'Yl.BO,B l ,B 2) des-.
Proposition 3.2: The set of coefficients
cribe a linear rational expectations equillbrium with inside information such
that
0*
x~ (p)
0*
°
°
0*
= YO + Y1P, B~ (p) = pe~ + B~ - px~ (p), i
~.
and
E
[0,1]
..
-19-
and
22
Yl a· I °rOy
04 = 1 2
2
or + 0y
Furthennore,
2
o<
Proof:
~l <
or
2
2 and
or + 0y
.
...
See Appendix I.
Corollary 3.2:
The set of coefficients
(to'tl,t2'PO'Pl'P2'~O'~1'~2)
describe a rational expectations equilibrium with public information such that
p{y,eI+e u )
= to
+ tlY + t2{e I +e u)
X~(y,p) = Po + PlY + P2 P, B~{y,P) = pe~
- and
and
+
B~ - PX~{y,p) i e[o, 1]
-20-
and llO. III
Proof:
and ll2 are as in Corollary 3.1.
Directly follows the proof of Proposition 3.2 by noting that price
yields no additional information to traders when y is public.
-214.
Equilibrium Outcomes
Given the simplicity of the equilibrium strategies, the main interest for
the results of Proposition 3.1 lies in the predicted equilibrium outcomes.
In
particular, one would like to know how the existence of an insider with market
power affects the information content of price and the responsiveness of price
to new information compared to the case when all traders behave competitively.
Alternatively, it is of interest to determine whether strategic insider trading
leads to a loss of informational efficiency by making the market IInoisier from
li
the point of view of each uninformed trader.
Yet the complexity of the expres-
sions in Propositions 3.1 and 3.2--which arises because the coefficients of
equilibrium price are jointly determined with the coefficients of uninformed
traders' asset demands--precludes a direct analysis.
However, some special
cases yield determinate results.
The first case occurs when there is no uncertainty about the initial wealth
of the different types of traders.
sitions 3.1 and 3.2 as Pl
Denoting the equilibrium prices of Propo-
and P2' the next example describes the relevant
equilibria when eI , e~ and e~ are fixed and known by all traders.
Example 4.1:
(0
2 = 0 2 = 0)
u
I
whereas
This example illustrates the well-known result by Grossman [1976], and
Grossman and Stiglitz [1978] that the competitive rational expectations equi1ib-
-22rium
pri~e
fully reveals all information in the market. Yet, under the condi-
tions of Example 4.1, this occurs even if the insider attempts to optimally exploit his market power.
In both cases, price varies one-to-one with informa-
tion, the coefficients wl and al reflecting the informativeness of the signal y. The only difference between the equilibrium outcomes of Propositions
3.1 and 3.2 are that
~O
= 1/26 0 and wl = 1/26 1: the insider who recognizes
his market power prevents price from being pushed up against himself with his
information.
In the absence of opportunities for re-trading, the informed
trader can exploit his market power to earn positive profits even when the equilibrium price perfectly reveals his information.
On the other hand, introducing even a small amount of noise in the form of
uncertainty about inftial wealth alters these conclusions. To illustrate the
consequences of non-competitive behavior in this situation, it is useful to
consider the cases when the jnsider has perfect information or is risk-neutral.
For, not only are the equilibrium outcomes under these assumptions easy to
characterize, but they are of interest in their right. In particular, the
assumption of risk neutrality seems appropriate for an informed trader with a
large initial position.
In Examples 4.2(a) and 4.3(a),
Pl and P2 again denote the equilibrium
prices of Propositions 3.1 and 3.2. However, for future reference, Examples
-4.2(b) and 4.3(b) describe the equilibria of Corollaries 3.1 and 3.2 where the
signal y is publicly observed.
-
In this case, the relevant prices are denoted
by Pl and P2"
Example 4.2:
(a~
=
0)
(a) y is private information
.....
-23and
P2 = Y
(b) y is public information
and
Example 4.3:
(a l
= 0)
(a) y is private information
and
2
2m
or
°v r
P2 = 2 + 2 + 2
2Y
or °v or + °v
(b) y is public information
2m
2
or
°v r
Pl = 2
2+ 2
2Y
or + °v or + 0v
22
au 0vor
eu
( 2 + 2)
or °v
and
Recall from Proposition 3.2 that the informed trader who behaves competitively reacts to his signal with a weight reflecting his risk aversion and the
precision of his signal, i.e.,
-24-
X1(y,p)
= BO
+
1
2y -
aI °v
When either ar
=0
or o~
(or2 + °v2)
22
p.
°rov aI
= 0,
his reaction to his information becomes
so strong that the effect of random supplies on price disappears
(~2
-0) and
price becomes fully revealing.
This is demonstrated by the expressions for P2
1n Examples 4.2(a) and 4.3(b).
In both cases, price varies one-to-one with the
signal y.
As 1n the case with fixed endowments, any
gain~
accruing to superior
information are eliminated in equilibrium. Further, the competitive outcome
achieves informational efficiency because the price of the riskY asset equals
the expected value of the future payoff r, conditional on observing the noisy
signal, i.e.,
2m
= E{rly) =
~r
,.
2
or
(4.1)
P2
When o~
= 0, this expression simplies to the value for P2 given in Example
2 y.
22+ 2
or + 0v or + °v
4.3{a), namely y.
On the other hand, if the insider can behave strategically and set prices,
equilibrium outcomes differ markedly from those emerging with competitive
behavior. As. in Example 4.1, only half as much information is incorporated in
price, i.e., nl
= ~ ~l' But, more important, price also fluctuates with the
aggregate endowment of uninformed traders, i.e.,
This last condition is always satisfied in equilibrium because uninformed .'
traders are assumed to be risk averse.
If, in addition, there is some hetero-
geneity among uninformed traders so that m ~ 1, price fails to be fully
revealing from the point of view of each such trader who observes the quantity
eI + me~or e I + (l-m)e~ but not e l or eu separately.
~
-25Furthermore, because the price of the risky asset is no longer an unbiased
"
"
"
predictor of future payoffs, the insider always earns positive profits in equilibrium. Using the expressions for price p and eI - XI implied by Examples
4.2 and 4.3 expected profits Ire given by
since Ql
<
O.
by the insider.
Notice that expected profits reflect the informa"tion possessed
Consequently, unlike the informed trader who behaves compe-
titively, the price-setting insider can benefit from his informational monopoly
as long as uninformed traders desire to share risks through the market but are
unsure about the quantity of shares held by the different types of traders. On
the other hand, with competitive behavior, p
= E(rly) so that expected profits
are always zero.
The above examples can also be used to examine the ro1e"of informational
advantage.
For this purpose, consider a risk-neutral monopolist who has no
private source of information.
From Corollary 3.1 and Example 4.2{b), his
expected profits can be expressed asS
-26Comparing (4.3) with (4.2) shows that when information is public, any
profit opportunities due to information are exhausted by the trades of the competitive fringe.
Since P2
Hence, (4.3) is positive but independent of the signal y.
dete~ines
uninformed traders' elasticity of demand for the risky
asset, this expression also shows that the large trader benefits from his
ability to set price under the same conditions as a textbook monopolist facing
an exogenous demand curve. This occurs when the demand function of competitive
traders is a negative function of price.
Alternatively, Example 4.3(b) shows that the actions of a risk-neutral
monopolist can prevent price from being an·unbiased predictor of future payoffs,
even when all information is public.
However, this example also shows that the
existence of such a trader does not affect
is incorporated into price:
th~efficienc;wi~h
which information
in both Examples 4.2(b) and 4'.3(b), the'~~'oefficient
of y in the expression for the price set by the
-
lar~e
trader, -Pl' is equal to
coefficient of y in the competitive price P2' But this is just a special case
of the results in Corollaries 3.1 and 3.2 that with public information, the coefficient of y merely reflects the informativeness of this signal, regardless of
how the large trader behaves.
Table 4.1 provides a more general comparison of the equilibrium outcomes
implied by Propositions 3.1 and 3.2. Given the simultaneous determination of the
c~efficients
of equilibrium prices, on the one hand, and the coefficients of
traders' asset demands, on the other, such a comparison cannot be achieved
analytically. Consequently, I numerically compute t~e rele~ant equ.i1ibria for
.
2
2
2
2
g1ven values of the exogenous parameters au, 01' or' 0v' au and aI' as
described in Appendix II. This determines the coefficients nO' n"
n2 and
n3 characterizing the equilibrium price in Proposition 3.1. The coefficients
~O'
~l
and
~2
of Proposition 3.2 are similarly determined. The expressions
-27-
in columns four and five of Tab1e
4.~
are then evaluated by using these coef-
ficients as
(4.4)
and
(4.5)
o~
"[2 =
o.
0
Table 4.1
~l
'lfl -
'lf2 - 62
'lf3 - ~2
2
2
(11 - °2
2
°1
<
0
>
0
<
0
2
0'1
<
0
>
0
<
0
2
<
0
>
0
<
0
>
°v
<
0
>
0
<
0
>
au
<
0
>
0
<
0
a1
<
0
>
0
O'r
2
?
"[1 - "[2
<
0
<
O.
0
<
0
0
<
0
>00
<
0
?
<
0
0
>
?
Table 4.1 compares the magnitudes of wl versus 61, 62 versus
0
~2'
etc.,
-28as the values of the exogenous parameters are individually varied.
the entries of
1<0 1
For example,
in column one of Table 4.1 show that, regardless of which
parameter is increased, the coefficient of the signal y in the price-setting
rule,
is always less than the corresponding coefficient in the competitively
~l'
determined price,
01.
Consequently, the first column replicates the results of
Examples 4.1, 4.2(a) and 4.2(b) and shows that price responds less to new information when the insider can behave strategically.
In this case, the insider profits
from his informational advantage by reducing the covariance of price with his
information.
The second and third columns compare the responsiveness of price under the
two equilibria to changes in the initial endowments of the insider and the uninformed traders, respectively.
Recall that ~~2'
02 an~ ~3 are all negative.
Consequently, columns two and three show that, compared" to the competitively
outcome, the price set by the insider fluctuates more with changes in the endowments of uninformed traders than with changes in his own endowment e I . In other
words, for most parameter values, 1~31 > 1°2 1 while 1~21 < 1°2 1: prices fall
less with increases in e I in the latter case because the insider acccounts for
the fact that a higher price increases the value of his initial endowments and
consequently his initial wealth.
The fourth column compares the absolute variability of price under the two
equilibria:
for most parameter values, the variance of price is greater in the
price-setting equilibrium. This result formalizes the intuition that strategic
trading may lead to lIexcess vo1ati1ity in asset prices and cause the market to
ll
be IInoisier from the point of view of small uninformed traders.
ll
However, Table
4.1 also shows that large variations in price may not prove very useful for
indicating the presence of an insider who manipulates price.
the second and sixth entries of columns four that
oi
For notice from
may be less than o~ if
the informed trader is very risk-averse (large aI) or suffers large vari-
-29-
ability in his initial we.a1th (large o~).
In any case, the last column of
Table 4.1 shows that if the information content of price is measured by the
proportion of variance in price attributed to the signal y,
then strategic
insider trading always causes equilibrium prices to be a worse indicator of the
information in the market compared to the competitive outcome.
Finally, for the sake of completeness, Tables 4.2 and 4.3 show comparative
statics results for the individual equilibria of Propositions 3.1 and 3.2.
A~
an example, notice that an entry of (-) in the first row, first column of Table
4.2 implies a fall in TIl
2
.
as au lncreases.
Under this interpretation, the
first columns of both tables show that the responsiveness of price to new informa-
(0;
increases) or the informed
·~increases)..
On the other hand, greater
tion decreases as the market becomes noisier
trader becomes more risk-averse
(a l
variability in the payoffs on the risky asset (large o~) or increases in the
risk aversion of uninformed traders (large av) have the opposite effect. Notice
also that an increase in the risk aversion of the informed trader does not necessari1y increase the overall variability of price when the insider behaves
strategically.
This is a reflection of the results in Table 4.1.
Another
interesting result is that greater risk aversion on the part of uninformed traders does not reduce the informativeness of the competitive price, i.e., '2 does
not change when au increases.
With strategic insider trading, a rise in this
parameter increases the overall variability of price, causing '1
to fall.
This
occurs because the coefficients on e l and eu increase in absolute value more
than the coefficient on the signal. When uninformed traders are very risk-averse,
their asset demands will not fluctuate very much with price.
Taking this reaction
into account, the insider will set price to more fully reflect changes in endowments and consequently ensure that his own position in the market,
more with information rather than endowments.
XI'
varies
-30-
Table 4.2
Compa ra t i ve Statics for Propositions 2.1
11T21
1T1
11T 31
2
01
'1"1
la11
2
au
+
+
2
01
+
+
2
or
+
+
2
°v
+
+
?
+
+
au
+
+
+
+
?
?
~
+
al
~
+
Table 4.3
Compa ra t ; ve Statics for Propositions 2.2
°1
1°2 1
2
02
"'2
ly11
2
au
+
+
021
+
+
a2
r
+
a2
al
+
+
+
V
au
+
+
+
+
+
+
+
unchanged
+
-315.
Some Empirical Implications
Summarizing, the previous sections described the equilibria in a simple
securities market model under different information structures, and behavioral
assumptions for individual traders.
In particular, this analysis emphasized the
consequences of strategic insider trading for the IInoisiness or informational
ll
efficiency of securities markets.
The analysis of these sections may also be
used to answer a question with potential implications for policy:
that is, is it
possible to distinguish the different equilibria and to detect the existence of
an insider who behaves strategically using observations on prices and quantities
traded?
Section 4 showed that focusing on such measures as the volatility of
~.
share prices may not be useful for this purpose because an informed trader who
tries to manipulate prices may actually reduce the variance of overall prices.
On the other hand, Examples 4.1-4.3 suggest that zero restrictions on equilibrium prices and cross-equation restrictions between the demand schedules of
uninformed traders and prices may help to distinguish the different equilibrium
configurations.
For example, when information is perfect
(a~ = 0), p; E(rly)
only in the presence of an insider who behaves strategically.
In this situa-
tion, price also depends on the in;"tial wealth of uninformed traders'.
ever, regardless of the information structure, the
equil~orium
How-
with strategic
trading will differ from the competitive outcome in that the price will depend on the initial holdings of uninformed traders as well as on the signal y.
On the other hand, cross-equation restrictions between prices and uninformed
traders' demands may be used to distinguish whether the information is lIinside
information or publicly known.
ll
Using the results of Example 4.3, Proposition
3.1 and its corollary, I obtain, with inside information, that
-321
where '12 = ~
(5.1 )
When y is publicly known but a large trader sets prices,
-Pl
= ~O
{5.Z}
_ 1
+ ~lY + ~Zeu where n2 -lJZ
Hence, the cross-equation restriction between '1Z and al' on the one
~Z
hand, and
and lJZ' on the other, may serve to identify the nature of
information by market participants. Under the more likely situation that the
large trader.Js risk-neutral
an unbiased predictor
given.
(al=O), Example 4.3 shows that price
will be
1
of4~uture
~
payoffs only if the
lar~
trader takes prices as
If the signal y is publicly known, it will be efficiently incorporated
into prices even in the presence of a dominant trader who sets prices.
More generally, the existence of a "wealth effect" distinguishes the equilibria where the large trader sets prices from those in which he behaves come
petitively. Adding and subtracting the term
~ Z Z 2 to the
al{Z - alalorov/(or + 0v})
expression for equilibrium" price in Proposition 3.1 and re-arranging terms yields
(5.3)
where
With al
<
0,
aZ+
> 0,
while '13
<
0,
el enters with a positive coefficient
because a higher price makes the "init.ial holdings of the. insider .more valuable.
With competitive behavior, e l and eu always enter with the same coefficient,
-33i.e. t from Proposition 3.2
(ptYteI + eu)
= ~O
+ ~lY + ~2(eI + eu}·
As a concluding remark t it should be pointed out that the above discussion
is highly suggestive.
Any serious empirical implementation would consider the
institutional framework as well as the nature of the observable series.
-34Appendix I
Proof of Proposition 3.1:
(i) Each uninformed trader conjectures that the
equilibrium pricing function is given by p(y,el,e u) = ~O + ~lY + ~2el + ~3eu'
Given this conjecture, the solution of each uninformed trader's problem can be
obtained as a special case of Hellwig [1980], pp. 483, equations (6a)-(6e).
This implies the demand function X~(p)
aO and al
as given in the text.
= aO
+ alP
for all
The solution for B~(P)
residual from the budget constraint in (3.1).
i E [O,lJ, with
is obtained as a
With constant absolute risk
aversion utility functions, the asset demands of individual traders do not
depend on initial wealth or endowments.
.~
(ii) Similarly, the insider conjectures that X~(p)
i E [O,lJ.
~iven
= aO
+ alP for all
this conjecture about the demand functions of the uninformed,
he uses the two constraints
to express El[-exp(-alWI)J solely in terms of p. This yields the problem
m~x E(r/y,eI ,eu)(eI + eu - aO - alP) + pel + BI - p(e + eu - aO - alP)
l
(1. 3)
- aI I2 Var(r/y,eI,eU)(e I + e u - a O - alP)2
Substituting for
-35-
and
(1. 5)
~O'
yields the solution for
~l'
~2
and
~3
given in the text and further
shows that all traders' conjectures are confirmed in equilibrium.
The price of
the riskless asset is normalized to one and the equilibrium holdings of the
riskless asset BI
for the informed trader are obtained as a residual from the
budget constraint (1.1) once XI
is known.
(iii) To complete the proof, I need to show that a solution exists for the
~
.
system of non-linear equations defining
~O'
and ~3 are independent of ~O and ~2
= -alo~~l'
and
...
needs to be found.
O.
Further, provided au
~3
~l'
~2
and
~3'
Since
~l'
only a solution for ~l
With constant absolute risk aversion utility func-
tions, the risky asset is not a Giffen good (see Grossman [1978]), i.e.,
Q1
<
~
0,
Ql
> -~.
Hence, whenever or2
~
0,
2
o ~ ~l
~
or
2
2
2(ov + or)
2 2
a1oyo r
2( 2 +
°v
2)
or
To show that a solution exists for
Also define
~2
~l
and
define the mappings
-36-
2
1
Tl (·,·)
T2
1 (. , .)
=
O'r
2
2 (O'r +
0')
22
-a I O'yO'r
= --i--";;'-;';~
2(
·f
'
2
O'r2 + O'2)
y
+-
Tl(
o .,. )
>
1
2a l
and
otherwise
·~··then, T(wl,w3) = (T~(wl,w3),T~(wl,w3)) is a continuous mapping which maps the
compact, convex set
s
0'
2
into itself.
wi
=0
= O.
0' 0'
Therefore, by Brouwer's fixed point theorem, T(wl,w3)
wi
fixed point wi and w3' with
aI
2 2
-a
r + -21-,OJ
= [0 , 2 +r 0'2) J x [ 2( 2I +y 2)
al
2 (O'y
r
O'r O'y
2
or O'y
= 0,
and
has a
Wo
defined from wi and w3 . When
rri and w3 clearly fallon the boundary of S, with
Otherwise, it can be shown by means of a contradiction that the
point is strictly interior to S,
Proof of Proposition 2.2:
as claimed.
(i) In this case, both types of traders con-
p(y,e l + e u) = 00 + 0lY + 02(e u + el). GiYen this
conjecture, the coefficients in the asset demands for both types of traders,
jecture a pricing function
i.e.,
eO'
el ,
S2 and YO and Yl
as defined in the text, are again found
as special cases of Hellwig [1980], pp. 483, equations (6a)-(6e).
Then, 00'
-37-
01
and 02 are determined from the market-clearing condition which specifies
that
In this case, it is easy to show that
and
where 00'
01
and 02 are defined in the statement of Proposition 2.2.
(ii) The existence of a solution for this set of equations follows exactly
as in part (iii) of the proof of Proposition 3.1.
solution for 01 and 02·
00 is determined from the
-38-
Appendix II
This appendix describes the algorithm for numerically computing the equilibria of Propositions 3.1 and 3.2, given a set of values for the exogenous
2 2 2 2
parameters or' 0v' 01' au, a I and au· For both cases, the coefficients of
uninformed traders' demand functions must be jointly determined with the coefficients of the equilibrium pricing function.
(i) For Proposition 3.2, the algorithm for jointly computing YO' Yl
and
01' 02 is simple: given a feasible initial ya1ue for 01' say o~, o~
1 021 and the remaining exogenous
= -a 10y2o11 and Y11 is determined from 01'
parameters as
00'
222
2
222
1 °lor - 0l(Oy + or} - 02(01 + au}
Yl =
2(~2 2 + ~2( 2 + 2}}
auo r U1 0y u2 01 au
1
Once Y1
is known,
oi
is determined from
2
or
Continuing in this way, an equilibrium is found for a given set of parameters
h
. h
..
when 1Y1i+1 - Y1i l < € were
€
1S C osen a prlorl.
(ii) For Proposition 2.1, this algorithm includes an additional step.
Again, assume an initial feasible
depends on a1
stituting for
and
~3
~1
but al
1
~1'
compute
depends on
in the expression for a1
~2
aal5 + ba 41 + Cal3 + da12 + ea1 + Vu
~1'
1
~2
2 1
= -aIoY~l.
~2
and n3·
Now
Hence, sub-
yields a fifth degree polynomial
= 0•
-39-
Given the values
1
~1'
1
~2
and the exogenous parameters, a numerical root
finding routine yields the roots of the above polynomial. The real negative
root, which always exists, is chosen as the value of a~. Then,
~
2
2_
or
1 -
-.2.....----.2:;-----..,1r-'--21'r'"J102r-""")I2r-~2:---'·
(or + °v}(2 - a 1a1or ov/(or + 0v)}
Continuing the above until
, ~li+1 -
~1i, <
€
. 1ds the solution.
Yle
-40References
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Inferences
From the Profitability of Insider Trading", Journal of Financial and
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~
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4
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-43-
Footnotes
1.
I thank Rick Green, Steven Matthews and Tom Palfrey for helpful comments.
Special thanks goes to Robert Miller and Robert Townsend for their
encouragement and advice.
2.
Remaining errors are my own.
These include Blume and Easley [1983J, Novshek and Sonnenschein [1982J
and Palfrey [1982J.
3.
The price of the riskless asset is normalized at one, yielding an
implicit risk-free interest rate of zero.
4.
The insider's holdings of the riskless asset BI are determined from
the budget constraint in (3.2).
5.
A similar result is obtained by Grinb1att and Ross [1982J.